Past exam of the differential equations 2 course of the University of Oxford / 2023 / 1 / a

The characteristic equation of the linear ordinary differential equation is $r^2+2r+2=0$, with roots $-1\pm i$. A real pair of homogeneous solutions is thereforeTheir Wronskian, with the convention $W=y_1{y}_2^{\prime }-y_1^{\prime }{y}_2$, isThus the two functions are linearly independent throughout the interval and span its homogeneous solutions.

Use the unweighted inner product $⟨w,y⟩={\int }_0^a\overline{w}ydx$. Two applications of integration by parts giveThe given boundary conditions set $y=0$ at both ends, but leave $y^{\prime }(0)$ and $y^{\prime }(a)$ independently variable. The remaining boundary term vanishes for every admissible $y$ precisely when $w(0)=w(a)=0$. Consequently the formal adjoint and the boundary conditions specifying the adjoint operator are